https://github.com/csirmaz/submodular-functions-6
Supplemental code for the paper "Attempting the impossible: enumerating extremal submodular functions for n=6"
https://github.com/csirmaz/submodular-functions-6
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Supplemental code for the paper "Attempting the impossible: enumerating extremal submodular functions for n=6"
- Host: GitHub
- URL: https://github.com/csirmaz/submodular-functions-6
- Owner: csirmaz
- License: gpl-3.0
- Created: 2024-10-18T23:17:51.000Z (7 months ago)
- Default Branch: main
- Last Pushed: 2024-12-30T17:04:19.000Z (5 months ago)
- Last Synced: 2024-12-30T18:19:10.895Z (5 months ago)
- Language: C
- Size: 165 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Enumerating extremal submodular functions
This repository contains
supplementary material for the paper:
E. P. Csirmaz and L. Csirmaz:
Attempting the impossible: enumerating extremal submodular functions for n=6
(in Mathematics, 2025).
The paper is available at https://www.mdpi.com/2227-7390/13/1/97 .
The preprint is available on arXiv at https://arxiv.org/abs/2410.15502 .
## Results for N=6
A partial list of 360 billion extremal submodular functions for N=6 is available at
https://zenodo.org/records/13954788 .
## Quickstart
```bash
# Clone the repository
$ git clone https://github.com/csirmaz/submodular-functions-6.git
$ cd submodular-functions-6
# Compile the binaries
$ sudo apt-get install gcc
$ ./build.sh
# Example 1: Compute all extremal rays for N=4 using recursive order
$ perl base.pl 4 --purge --rec /tmp/ex4
# The results will be written to /tmp/ex4-24.txt
# Example 2: Generate all extremal rays adjacent to the Vamos-ray defined
# in vamos.txt
$ perl runjob.pl 4 --ray vamos.txt /tmp/v
# The results will be written to /tmp/v-all.txt
# Example 3: Continuing from the known N=6 rays in Zenodo, keep searching
# for neighboring rays. Download the latest rays:
$ wget https://zenodo.org/records/13954788/files/6rays-26.txt?download=1 -O 6rays-26.txt
# Generate all neighboring rays of each listed ray:
$ perl runjob.pl 6 6rays-26.txt /tmp/out
# The results will be written to /tmp/out-all.txt in a
# ": " format.
```
## Files
`ax.c` - p-standardized submodular inequalities in C structure for base set
sizes N=3,4,5,6. Human-readable versions of the inequalities are included.
`sym.c` - Conversion matrices for computing permutations and the dual, as
C structures.
`bitmap.c`, `bitmap.h` - Implementation of bitmap-manipulating routines as required by
the DD method. It uses assembler instructions to count the bits in a 64-bit word.
`iterate.c` - Implementation of a single iteration of the DD method as a pipe.
It reads the description of the problem, the next inequality, and then
produces the result of the iteration. Should be compiled for each
base size N separately. Used by the controller program `base.pl`.
To compile for a given N, use
`gcc -O3 -o bin/iter -D RANK= iterate.c bitmap.c`
`neig.c` - A version of `iterate.c` for computing the neighbors of a ray. Used by
the controller program `runjob.pl`. Should be compiled for each base
size N separately. To compile for a given N, use
`gcc -O3 -o bin/neig -D RANK= neig.c bitmap.c`
`newrays.c` - Generates the extremal rays from the neighbor enumeration. It can
produce a canonical representative from the corresponding orbit
rather than the ray itself. Used by the controller program `runjob.pl`.
Sould be compiled for each base size N separately. To compile for a given N, use
`gcc -O3 -o bin/newr -D RANK= newrays.c`
`extract.c` - Generate extremal rays from an intermediate cone of the DD method
as produced by the controller program `base.pl`. It can produce a
canonical representative from the corresponding orbit rather than
the ray itself. Should be compiled for each base size N separately.
To compile for a given N, use
`gcc -O3 -o bin/extr -D RANK= extract.c bitmap.c`
`base.pl` - Controller program to execute all iterations of the DD method for
the submodular cone. Arguments are the base size and the ordering
method. Generates the initial approximation, then calls `bin/iter`
until the algorithm terminates. When stopped, the last result can be
used as input to `bin/extr`.
`runjob.pl` - Controller program to compute rays adjacent to a list of extremal
rays. Arguments are the base size, ordering method, and the list of
extremal rays. For each ray generates the initial approximation,
and calls `bin/neig` until the DD method completes. Finally calls
`bin/newr` to generate the neighbors or their canonical representatives.
`vamos.txt` - One of the six Vamos extremal rays of the 4-element submodular
cone.
`build.sh` - Script to call `gcc` to build all binaries.
### Raw data
`figures-data` - Raw data for the figures and charts in the paper can be found
in the folder https://github.com/csirmaz/submodular-functions-6/tree/main/figures-data .